After doing some numerical simulations, I rediscovered that the Bell numbers are periodic modulo $n$, that is to say we have the following identities :
\begin{align} B_{n+3} &= B_n\mod{2} \\\\ B_{n+13} &= B_n\mod{3} \\\\ B_{n+12} &= B_n\mod{4} \\\\ B_{n+781} &= B_n\mod{5} \\\\ &\vdots \end{align}
In fact, I realized that there is a Wikipedia chapter that talks about this and an a OEIS sequence for all the periods modulo $n$. The Wikipedia chapter claims that for a prime $p$ the period modulo $p$ must divide $(p^p-1)/(p-1)$
Questions :
- How can we prove such identities and their generalizations to higher modulos ?
- How can we indeed prove that the Bell numbers are periodic modulo $p$ with a period that divides $(p^p-1)/(p-1)$ ?